Solution 1 (c).

See text and/or instructor's solution manual.

Answer.      valid for  .  

Solution.   Given  ,  the derivatives are  

                    ,    ,    ,    ,    etc.  

In general      for   .

Evaluate the derivative at and get  ,  and

                    

      By Theorem 7.4, the coefficients of the Maclaurin series are

                         for   ,  

and the sequence of coefficients is

                      

Hence, the Maclaurin series is  

                    

                      

We are done.   

                    The images of the disk    under   ,  for  .  

                    The image of the disk    under the mapping   .  

We are really done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell